Arithmetic and Geometric Sequences

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What is an Arithmetic Sequence?

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An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. For example, {2, 5, 8, 11} is an arithmetic sequence with a common difference of 3.

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What is a Geometric Sequence?

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A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, {3, 6, 12, 24} is a geometric sequence with a common ratio of 2.

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How do you find the common difference in an Arithmetic Sequence?

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The common difference in an arithmetic sequence can be found by subtracting any term from the subsequent term. For example, in the sequence {4, 7, 10}, the common difference is 7 - 4 = 3.

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How do you find the common ratio in a Geometric Sequence?

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The common ratio in a geometric sequence can be found by dividing any term by the previous term. For example, in the sequence {2, 6, 18}, the common ratio is 6 / 2 = 3.

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What is the formula for the nth term of an Arithmetic Sequence?

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The nth term of an arithmetic sequence can be calculated using the formula: a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference.

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What is the formula for the nth term of a Geometric Sequence?

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The nth term of a geometric sequence can be calculated using the formula: a_n = a_1 * r^(n - 1), where a_1 is the first term and r is the common ratio.

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Given the sequence {5, 10, 15, ...}, identify if it is Arithmetic or Geometric.

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The sequence {5, 10, 15, ...} is an Arithmetic sequence because the difference between consecutive terms is constant (5).

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Given the sequence {2, 4, 8, ...}, identify if it is Arithmetic or Geometric.

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The sequence {2, 4, 8, ...} is a Geometric sequence because each term is multiplied by 2 to get the next term.

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What are the first three terms of the sequence defined by a_n = 3 + (n - 1)2?

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The first three terms are 3, 5, 7.

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If the 1st term of an arithmetic sequence is 10 and the common difference is 4, what is the 5th term?

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The 5th term is 10 + (5 - 1)4 = 10 + 16 = 26.